Digital Object Identifier (DOI) 10.1007/s00205-003-0268-3 Arch. Rational Mech. Anal. 169 (2003) 265–304 Relaxation of Hamilton-Jacobi Equations Hitoshi Ishii & Paola Loreti Communicated by P.-L. Lions Abstract We study the relaxation of Hamilton-Jacobi equations. The relaxation in our terminology is the following phenomenon: the pointwise supremum over a certain collection of subsolutions, in the almost everywhere sense, of a Hamilton-Jacobi equation yields a viscosity solution of the “convexified” Hamilton-Jacobi equa- tion. This phenomenon has recently been observed in [13] in eikonal equations. We show in this paper that this relaxation is a common phenomenon for a wide range of Hamilton-Jacobi equations. 1. Introduction In this paper we study the Hamilton-Jacobi equation H (x, u(x), Du(x)) = 0 in , (1.1) where  is an open subset of R n , H is a given real-valued function on  × R × R n , and u is a real-valued unknown function on , and we are interested in an obser- vation concerning (1.1) in [13] and its generalization, which we call the relaxation of Hamilton-Jacobi equations. This observation in [13] is stated as follows. Let H ∈ C(R n ) be a function satisfying H (p) > 0 for p ∈ R n \{0}, H (λp) = λH(p) for (λ, p) ∈[0, ∞) × R n . (1.2) Let  H denote the convex envelope of the function H . Consider the eikonal equation  H(Du(x)) = 1 in  (1.3)